Reduction of Lie–Jordan Banach algebras and quantum states

Abstract

Abstract. A theory of reduction of Lie-Jordan Banach algebras with respect to either a Jordan ideal or a Lie-Jordan subalgebra is presented. This theory is compared with the standard reduction of C * -algebras of observables of a quantum system in the presence of quantum constraints. It is shown that the later corresponds to the particular instance of the reduction of Lie-Jordan Banach algebras with respect to a Lie-Jordan subalgebra as described in this paper. The space of states of the reduced Lie-Jordan Banach algebras is described in terms of equivalence classes of extensions to the full algebra and their GNS representations are characterized in the same way. A few simple examples are discussed that illustrates some of the main results.

Schwinger’s picture of quantum mechanics III: The statistical interpretation

“…States are just normalised positive linear functionals on the C * -algebra of the groupoid, hence, they are blind to the specific details of the algebraic structure of the algebra (they just preserve the positive cone of the algebra). It is true though that the C * -algebra structure can be recovered from the space of states, more precisely, because of Kadison's theorem [25], the real part of a C * -algebra is isometrically isomorphic to the space of all w * -continuous affine functions on its state space, and then, as it was shown by Falceto et al, the C * -algebra can be constructed on the space of affine function on the state space iff such space has the structure of a Lie-Jordan-Banach algebra [26] (see also [27,28]).…”

“…that will be assumed in addition to the strict factorization property (26). Notice that condition (27) is independent of the factorization condition (26) and it can be lifted when dealing with open systems. Note also that as a consequence of the factorization condition, Eq.…”

Differential Geometry of Quantum States, Observables and Evolution

“…Because of the properties of the symmetric product on Hermitean operators, the bracket < ·, · > turns out to be a Jordan product. Furthermore, the set of observables endowed with the antisymmetric product {·, ·} and the symmetric product < ·, · > is a Lie-Jordan algebra [6,7,14]. By complexification, that is, considering complex-valued functions F A = f a 1 + ıf a 2 for some Hermitean a 1 , a 2 , we obtain a realization of the C * -algebra B(H) by means of smooth functions on P = CP(H) according to [8,11]:…”

Geometrical structures for classical and quantum probability spaces

“…Consequently, the vector space of linear functions endowed with the Lie product [[· , ·]] and the symmetric product ⊙ is a Lie-Jordan algebra. Finally, we may introduce an associative product on the complexification of the space of linear functions, which turns out to define a C * -algebra structure, setting: [11] f a ⋆ f b :…”